The finitistic dimension of rings with Morita context
type: Item
instance of: scholarly article
title: The finitistic dimension of rings with Morita context
zbMATH Open document ID: 0924.16008
author: Zhifeng Hao
published in: Southeast Asian Bulletin of Mathematics
publication date: 1999-09-20T00:00:00Z
review text: Let \(T=(T_{ij})\) be a \(2\times 2\) matrix ring, where \(T_{11}\), \(T_{22}\) are rings (associative and unitary), and \(T_{12}\), \(T_{21}\) are bimodules over these rings such that \((R,S,M,N)\) is a Morita context. The author assumes that all left modules are Artinian. Every left \(T\)-module is determined by a quadruple \((P,Q,f,g)\), where \(P\) is a left \(R\)-module, \(Q\) is a left \(S\)-module and \(f\colon M\otimes_SQ\to P\), \(g\colon N\otimes_RP\to Q\) are homomorphisms of modules making commutative certain diagrams. Proposition 3.1 gives a bound for the projective dimension of a left \(T\)-module: \[ \begin{multlined}\text{pd}(P,Q,f,g)\leq 1+\sup\{\text{pd}(_SQ),1+\text{pd}(_SN)+\text{pd}(_{R/(MN)}\ker(f)),\\ \text{pd}(_SN)+\text{pd}(_{R/(MN)}P/f(M\otimes_SQ))\},\end{multlined} \] where \(MN\subseteq R\) denotes one of the trace ideals. The left finitistic dimension \(\text{lfPD}(T)\) of \(T\), defined as the supremum of the projective dimensions of those finitely generated left modules having finite projective dimension, is then studied. For example, the author deduces, in the case that \(M_S\) is flat and \(_SN\) has finite projective dimension, that \(\text{lfPD}(T)\leq\text{lfPD}(S)+\text{LGD}(R/(MN))+2\), where \(\text{LGD}(R/(MN))\) is the left global dimension of \(R/(MN)\).
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1267192
zbMATH Keywords: Morita contexts
zbMATH Keywords: matrix rings
zbMATH Keywords: projective dimensions
zbMATH Keywords: finitistic dimensions
zbMATH Keywords: global dimensions
reviewed by: José Gómez-Torrecillas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1267192